G482 Module 4 - Wave Motion
Wave Basics
A wave carries energy from one place to another. Its constituent parts move around a
localised point, but do not travel with the wave on average. There are two types of wave:
Longitudinal:
And Transverse:
Longitudinal Waves
Sound is an example of a longitudinal wave. A speaker vibrates, which causes the air particles
next to it to be pushed away; these particles hit their neighbours, which hit their neighbours
and so on. This successive movement of particles allows
allows the vibration to travel away from the
speaker as a wave.. A wave that travels is called a progressive wave.. [Ripple tank, slinky &
wave machine demo, echalk wave laboratory]
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Longitudinal waves oscillate parallel to their direction of propagation.
propagation
A pushed slinky spring wave is a longitudinal wave.
This diagram shows where the wavelength is on a longitudinal wave, as you can see, the wave
is oscillating in the same direction as it is moving. Areas where the air molecules are packed
closely together are called compressions, and areas where they are spread out are called
rarefactions.
The amplitude of a sound wave is how
how far a molecule moves away from its equilibrium
position.
Transverse Waves
Transverse waves are slightly easier to visualise; they oscillate at right angles to their
direction of propagation.
Waves on a string are good examples of transverse waves, we can
can also create a transverse
wave on a slinky spring by moving it from side to side.
Although from the surface water waves look like transverse waves, they’re actually a mixture
of both transverse and longitudinal waves.
Wave Properties
Frequency, f
Frequency is the number of complete oscillations that pass a particular point every second.
Its unit is Hertz (Hz), which simply means ‘per second’
Period, T
Period is the time taken for one complete oscillation. It is measured in seconds (s)
T = 1/f
Where T = period in seconds (s)
f = frequency in Hertz (Hz)
Amplitude, A
This is the maximum displacement of the wave from its equilibrium position. The greater the
amplitude the greater the energy transfer
Wavelength, λ
This is the distance between 2 identical points on neighbouring waves.
Unit is the metre (m)
Phase Difference Ф
This is measured in degrees or radians and details the fraction of a wave which lies between
2 waves. Here a difference of one wave equals 360o or 2π radians.
The following diagrams show a phase difference:
In the diagram above:
A lags B by 900
this is the same as saying B leads A by 900
The waves are 900 (or π/2 radians) out of phase
In the diagram above
A lags B by 1800
or A leads B by 1800
In this case the waves are 1800 (or π radians) out of phase.
[CRO demo & pendulum demo to show phase difference]
Wave Speed
The speed of a wave can be calculated quite easily using the following equation:
v=fλ
Where
v
is the wave speed in metres per second (m/s)
f
is the frequency of the wave in Hertz (Hz)
λ
is the wavelength of the wave in metres (m)
Derivation
Velocity, v = displacement, s / time, t
but in a time T equal to the period of the wave, s will be λ
v = λ/T, but T also equals 1/f
so v = f λ
Example
Station Radio 4 on ‘long wave’ radio has a frequency of 198 kHz.
What is the wavelength of the waves that arrive at your radio?
The speed of radio waves (electromagnetic radiation) is 3.00 x 108ms-1
In the equation, the frequency must always be in hertz, not kilohertz.
198 kHz = 198 x 103 Hz
C=fλ
3.00 x 108 ms-1 = 198 x 103 Hz x λ
Therefore
3.00 × 108 ms −1
λ=
= 1520m
198 × 10 3 Hz
Questions on page 167 on the wave equation.
[CRO demo to prove the wave equation]
to 3 s.f. (that’s nearly a mile!)
Reflection
When a wave hits a solid surface it will reflect. The law of reflection is:
Angle of incidence = angle of reflection
The incident ray, normal and reflected ray all lie in the same plane.
All surfaces reflect some light, but only very smooth surfaces will reflect light uniformly
enough to form a reflected image. Rougher surfaces scatter the light which is why we can
see a wall but we don’t see our faces reflected from it (unless it’s very well polished!).
If light is shone onto a transparent medium, a small fraction of the light will always be
reflected.
The same law of reflection is true for sound waves. But it’s much more difficult to show this
in an experiment!
Refraction
This is the change in direction of travel of a wave that occurs at the interface between two
regions in which the wave speeds are different.
When waves slow down they refract towards the normal
[Software demo. Ray boxes and glass blocks experiment]
The frequency of the wave remains constant, the speed decreases and the wavelength
decreases.
Examples - the wave will refract towards the normal
light: air into glass
sound: water into air
Light slows down when it moves from air to glass, but you’ll remember that sound travels
faster through water than air so it slows down when it moves from water into air.
When waves speed up they refract away from the normal
The amount a wave bends when it moves between media (the plural of medium) depends on:
•
The wavelength of the incident wave (short wavelengths deviate most)
•
The change in speed which occurs (the greater the change in speed, the greater the
deviation)
To understand why refraction happens, imagine a line of soldiers marching across tarmac.
They encounter a sandy area at an angle to
their direction of march (see diagram).
As the soldier on the left gets to the sand he
slows down
wn (because it’s harder to walk over
sand than tarmac), the soldier next to him
can continue walking at a higher speed for a
short while though, until he too hits the sand.
This continues all along the line causing the
line to bend slightly, and the soldiers
soldie to
bunch up.
Diffraction
If a wave’s wavefronts pass through a gap
that is similar in size to its wavelength, the wavefronts will bend. This causes the wave to
spread out.
[ripple tank demo]
If the gap is considerably larger than the wavelength of the
the wave then the edges of the
wavefront bend and less spreading out occurs.
The parts of the wavefront that touch the walls either side of the gap are reflected back.
What’s a Wavefront?
A wavefront is simply a way of showing the wavelength of a progressive
progressive wave pictorially. It’s
useful when different parts of a wave encounter different objects, such as in the diagram
above. Each line on the wavefront diagram represents a peak on the progressive wave.
2.4.3 Electromagnetic Waves
Electromagnetic (EM) radiation can be produced by many different means, and have a very
large range of wavelengths (and therefore frequencies). Visible light is one example of
electromagnetic radiation, but it is only a tiny part of the Electromagnetic Spectrum.
The Electromagnetic Spectrum
Radio waves
Ronald
Micro waves
Macdonald
Infrared
radiation
Is
Light (visible)
Ultraviolet
radiation
X-radiation
Lubricated
Using
Gamma
radiation
Grease
Long
Wavelength
Low frequency
Short
wavelength
High
Frequency
eXtra
All EM radiation travels at the same speed in a vacuum. This speed is known as the speed of
light. We use the symbol c to represent this speed.
c = 299,798,458 ms-1
c ~ 3x 108 ms-1
Using this fact and the wave equation, fill in the frequencies in the table below.
Type of radiation
Radio (longwave)
Radio (UHF)
Microwave
Infrared
Visible light
Ultraviolet
X-Rays
Gamma
Wavelength / m
>106
10-1
10-1 to 10-3
10-3 to 7 x 10-7
7 x 10-7 to 4 x 10-7
4 x 10-7 to 10-8
10-8 to 10-13
10-10 to 10-16
Frequency / Hz
There is no clear division between different ranges in the spectrum; the divisions given here
are somewhat arbitrary. EM radiation is usually defined by the process that produces it.
Remember: all EM radiation behaves like visible light, which you are familiar with. It travels
in straight lines, can be reflected, refracted, diffracted and dispersed, and it all travels at
the same speed in a vacuum! Visible light is simply one particular wavelength of EM radiation.
[Research activity – how are the different types of EM radiation produced?]
Visible Light
(This is not going to photocopy well! Colour it in)
Colour
Red
Orange
Yellow
Green
Blue
Indigo
Violet
Mnemonic
Richard
Of
York
Gave
Battle
In
Vain
Wavelength (nm)
670 – 750
590 - 670
570 - 590
495 – 570
450 – 495
380 - 450
These wavelength ranges are just an approximation, because the spectrum does not have
clear cut lines dividing the colours. Also, indigo is a made up colour, devised by Isaac Newton
to give the spectrum seven colours. Newton was really into theology and seven was thought to
be a holy number.
Uses of EM Radiation
Electromagnetic radiation is really useful stuff, and we have a lot of uses for it. A major use
is in imaging, when we want to see really small things. If the object you want to see is smaller
than the wavelength of the radiation you’re using to look at it then you can’t get a clear
image. So we can’t see anything smaller than 400nm using our eyes, no matter how good an
optical microscope we use. Instead we have to use radiation with smaller wavelengths, such
as X-Rays to probe the structure of crystals (this is known as X-Ray crystallography).
Conversely, longer wavelengths tend to travel further, as they are less prone to
interference. This is why we use radio waves for long distance communication within the
Earth’s atmosphere.
Ultraviolet Light And Skin
We all know that going out on a sunny day can give you a suntan, or even sunburn and that
sunscreen is useful if you don’t fancy looking like a lobster for a week. It is actually
Ultraviolet (UV) radiation from the sun that causes these effects, and it comes in three
types:
UV – A
Is the long wave ultraviolet, also known as blacklight (the kind you find in night clubs). Its
long wavelength allows it to penetrate cloud cover, large amounts of atmosphere (such as
during sunrise and sunset), and deeply into your skin. Although most of the UV that falls on
Earth is UV-A, it has relatively low energy and is not as damaging as the other types. UV-A
can cause skin cancer and a suntan in very large quantities.
UV – B
Is a more damaging kind of UV. It inhabits a narrow band of the UV spectrum and is the main
cause of sunburn and skin cancer (and as a result, tanning). UV-B penetrates only the outer
layer of your skin and knocks molecules off your DNA and skin cells, causing damage. The
DNA damage can lead to cancer, and the skin cell damage gives you erythema (reddening of
the skin, sunburn). The passage of UV through your skin also stimulates melanocytes, which
produce a brown pigment called melanin. Having extra melanin in your skin scatters and
filters some of the UV, although even a strong tan is only equivalent to SPF 2 sunscreen.
UV-C
Is extremely damaging, but is almost completely filtered out by the ozone layer. This is why
the hole in the ozone layer was such big news 10 years ago. UV-C is used in industry to
sterilise things because it’s so damaging to living cells and can kill microbes very effectively.
Just imagine what it’d do to your skin if the atmosphere wasn’t there to protect you!
Sunscreen
Sunscreen contains chemicals that block UV-B. Causing the energy they transfer
to you to be dissipated as heat rather than knocking molecules off your cells and
DNA!
SPF (Sun Protection Factor) tells you how effective the sunscreen is at filtering
UV-B. It tells you how many times more protection (than none at all) the product
provides you with. So 15 minutes in the sun with SPF 15 is equivalent to 1 min in
the sun with no sunscreen.
You can buy sunscreen that protects against UV-A as well.
Polarisation
Transverse waves can vibrate in
all directions, right through
360o. This diagram shows a
transverse wave vibrating in
two directions, but normal light
vibrates through 360o.
Polarisation is the act of
forcing a transverse wave to
vibrate in only one direction.
Polaroid
Polaroid is a type of filter that polarises light. It works by only allowing through light that is
vibrating in one direction.
This diagram shows
the polarisation of
light. Initially it is
able to vibrate in
many directions, but
only light vibrating in
the vertical direction
can pass through the
vertical polarising filter. The vertically polarised light is then passed through a horizontally
oriented piece of Polaroid, which will not let the vertically polarised light pass. Nothing
emerges.
Reflections and Sunglasses
Reflected light is often partially plane polarised (in the same direction as the surface they
reflect from). This means that by passing the light through a vertically polarising filter (or
Polaroid sunglasses) you can drastically cut down the glare from reflections.
We can test this with a bit of glass, a light bulb and some polarising filters!
What will a polarising filter do to a photograph?
Notice how the reflection in the river is dulled down. Also notice that the sky is darker,
showing that a fair amount of the light coming from the sky is filtered out (because it’s
normally not polarised).
Intensity of Light
When a wave hitss a surface, it transmits some energy to that surface. The amount of energy
(in Joules) per second that a wave can transfer is known as its power,, just like in any other
situation in which energy is transferred.
It is also useful to talk about the intensity of a wave, that is how much energy it transfers
to each square metre it falls on. Intensity is given by:
Intensity (Wm-2) = power (W) /cross sectional area (m2).
A wave’s intensity is proportional to the square of its amplitude:
Intensity
amplitude2
So if the amplitude of a wave doubles, it transfers four times the power to every square
metre of target that it hits.
Malus’ Law
Malus’ law tells us how much the intensity of polarised light drops after it has passed
through a second polarising filter.
Where
‫ ܫ‬ൌ ‫ܫ‬଴ ܿ‫ ݏ݋‬ଶ ߠ
I is the intensity of the transmitted beam
I0 is the intensity of the incident beam
θ is the angle the polarising filter makes with the polarised light
Example
A sheet of Polaroid is being used to reduce the
the intensity of a beam of polarised light. What
angle should the transmission axis of the Polaroid make with the plane of polarisation of the
beam in order to reduce the intensity of the beam by 50%?
Answer: Since we want the transmitted beam to be 50% of the incident beam, we replace
I with I0/2, then:
‫ܫ‬଴
ൌ ‫ܫ‬଴ ܿ‫ ݏ݋‬ଶ ߠ
2
1
ൌ ܿ‫ ݏ݋‬ଶ ߠ
2
1
ඨ ൌ ܿ‫ ݏ݋‬ଶ ߠ
2
ߠ ൌ 45௢
Interference – Principle of Superposition
When two groups of waves (called wave trains) meet and overlap they interfere with each
other. The resulting amplitude will depend on the amplitudes of both the waves at that point.
If the crest of one wave meets the crest of the other the waves are said to be in phase and
the resulting intensity will be large. This is known as constructive interference. If the crest
of one wave meets the trough of the other (and the waves are of equal amplitude) they are
said to be out of phase by π then the resulting intensity will be zero. This is known as
destructive interference.
This phase difference may be produced by allowing the two sets of waves to travel different
distances - this difference in distance of travel is called the path difference between the
two waves.
There will be many intermediate conditions between these two extremes that will give a
small variation in intensity but we will confine ourselves to total constructive or total
destructive interference for the moment.
The diagrams in Figure 1 below show two waves of equal amplitudes with different phase and
path differences between them. The first pair have a phase difference of π or 180o and a
path difference of an odd number of half-wavelengths. The second pair have a phase
+
(a) destructive
=
+
Figure 1
=
(b) constructive
difference of zero and a path difference of a whole number of wavelengths, including zero.
Figure 1(a) shows destructive interference and Figure 1(b) constructive interference.
[Exercise – Graphical techniques for determining superposition]
To obtain a static interference pattern at a point (that is, one that is constant with time) we
must have
(a)
two sources of the same wavelength, and
(b)
two sources which have a constant phase difference between them.
Sources with synchronised phase changes between them (i.e. they have the same wavelength
and speed) are called coherent sources and those with random phase changes are called
incoherent sources.
This condition is met by two speakers connected to a signal generator because the sound
waves that they emit are continuous – there are no breaks in the waves.
Interference between two waves
The diagrams shows two sources S1 and S2
Minimum – crest meets
emitting waves - they could be light, sound or
microwaves.
Maximum – crest meets
The plan view of the waves in Figure 3 shows
Minimum – crest meets
waves coming from two slits and interfering
S1
with each other.
Maximum – crest meets
S2
This type of arrangement is like that
Minimum – crest meets
produced in a ripple tank or in the double slits
experiment with light (see later).
Maximum – crest meets
Figure 3
Minimum – crest meets
It should be realised that between the maxima and minima the intensity will change gradually
from one extreme to the other.
Figure 4 (a)
Figure 4(a) shows light interfering as it passes
through two slits. In Figure 4(a) the appearance
of the interference pattern on a screen placed
in the path of the beam is shown.
You can see the maxima and minima and the way
in which the intensity changes from one to the
other.
Changing the wavelength of the light (its
wavelength), the separation of the slits or the
distance of the slits from the screen will all give
changes in the separation of the maxima in the
interference pattern.
Figure 4 (b)
Interference Experiment With Speakers
Figure 5 shows the interference effects of two speakers. The sound waves spread out all
round the speakers and a static interference
minimum
maximum
pattern is formed. (Not all the maxima and minima
are labelled). You can hear this by setting up two
speakers in the lab connected to one signal
maximum
generator and then simply walking round the room.
You will hear the sound go from loud to soft as you
Figure 5
pass from maximum to minimum. (A frequency of
around 400 Hz is suitable).
maximum
minimum
Young’s Double Slit Experiment
Up until the end of the eighteenth century physicists weren’t sure whether light behaved
like a wave, or like a stream of particles (corpuscles was the name Newton gave them). When
Thomas Young came along, he proved once and for all that light can be described as a wave by
causing two light sources to interfere with each other. There was no way that this effect
could be caused by a stream of particles and so the wave theory was proven. Since the
experiment is based on classical physics (rather than quantum, we’ll come to that later), this
experiment is a classical confirmation of the wave nature of light.
Later in the course we’ll see that light actually can be described as particles, but we won’t
worry too much about that yet!
This is how he did it:
single slit
0.5 – 1.0 m
source
double slit
screen
As you know, a coherent light source is required for interference. This is achieved using a
single light bulb and splitting the wavefronts in two using a double slit arrangement.
[Carry out Young’s Double slit experiment, measure fringe separation, distance from screen
and slit separation, calculate the wavelength of the light using the equation below.]
The fringe separation achieved using a particular slit separation, distance from a screen and
wavelength of light can be determined using:
ࢇ࢞
ࣅൌ
ࡰ
Where
λ is the wavelength of light (m)
a is the slit separation (m)
x is the fringe separation (m)
D is the distance from the slits to the screen (m)
Derivation
We’re looking to find λ, which is the difference between the path lengths of the two rays
when they meet at the first fringe (constructive interference). So it will be useful to find
their lengths first.
C
x
S1
X
a
A
Y
S2
D
Figure 2
The two rays form hypotenuses of right angled triangles: S1XC and S2YC. Therefore we can
determine their lengths using Pythagoras’ theorem:
S1C2 = S1X2 + XC2
S2C2 = S2Y2 + YC2
And since we know those distances:
S1C2 = D2 + (x – a/2)2
S2C2 = D2 + (x + a/2)2
So the difference between the length of the two rays is just one equation, minus the other:
S2C2 – S1C2 = (x + a/2)2 – (x – a/2)2
Each side of this equation is the difference of two squares. So we can write:
(S2C + S1C)(S2C – S1C) = (x + a/2 + x – a/2) (x + a/2 – x – a/2)
Which simplifies to:
(S2C + S1C)(S2C – S1C) = 2ax
Since S1C and S2C are so similar in length to D (to within 1 µm in 50cm) we can say that
S1C + S2C equals 2D. We also know that the path difference of the two rays must be equal to
one wavelength of the light λ, so Thus S1C – S2C = λ.
2D λ = 2ax
And
Diffraction Revisited
We can prove that diffraction happens because we can pick up an electromagnetic signal
(from light or microwaves) round the edge of a barrier. If the EM radiation travelled purely
in straight lines, we couldn’t do this.
[ripple tank, light box and single slit & 3cm wave kit demo]
Remember that the gap that the wave travels through needs to be comparable in size to its
wavelength for diffraction to happen most effectively.
Diffraction of Light
We can cause light to diffract by passing it through
through a small enough slit.
[demo laser & single slit]
The laser is a source of monochromatic light (all light it emits has the same wavelength, and
is therefore the same colour). This means we can see it interfere with itself quite easily, no
other wavelengths
hs get in the way.
When we shine laser light onto a single slit we get this diffraction pattern:
This is because light coming from different
parts of the slit travels different distances.
Notice that the maxima are very wide and vary in intensity.
Multiple Slit Diffraction
If you increase the number of slits that
you shine light through, the maxima become
thinner and much sharper. This makes it
much easier to measure their position.
The top interference pattern has been
created using two slits, and the bottom one
using five.
Diffraction Gratings
These produce interference in a similar way to 2 slits. However, a diffraction grating has
many thousands of slits. These are produced from glass sheets which are coated with an
opaque (can’t be seen through) layer. They then have a large number of equally spaced
parallel lines scratched on to them or etched out of them to reveal areas of glass for light to
pass through.
Diffraction grating formula
d sinθ = n λ
Where
d
is the distance between the slits in metres (m)
θ
is the angle between the direction of the incident light and the
diffracted light in degrees (°)
n
is the order number
λ
is the wavelength in metres (m)
The diffracted light is visible several times at greater angles. Each time it reappears it is
said to have a greater order number.
This is why light appears split up when you look through a diffraction grating. Each of the
wavelengths that make up white light constructively interferes at a different angle,
spreading them out.
[Practical Exercise – measure the wavelength of the sodium orange pair and various LEDs]
Measuring the Wavelength of Light with a Diffraction Grating
It’s quite easy to measure the wavelength of monochromatic light using a diffraction grating:
•
Shine your light through a diffraction grating onto a screen;
•
Measure the distance from the diffraction grating to the screen;
•
Measure how far apart the first order fringe is from the central one;
•
Use a bit of trigonometry to work out the angle from the central fringe to the centre
of the grating to the first fringe (θ);
•
Use the information printed on the grating to work out how far apart the slits are (d);
•
Plug these numbers into the diffraction grating formula and calculate the wavelength
(λ);
•
Repeat for a second order fringe (and higher) to check your result.
Derivation
The diffraction grating is made up of many
thousands of slits, but as we have seen the
maxima are in the same place no matter how
many slits you use. This diagram shows the
light from six slits but it demonstrates the
principle nicely.
Note also that for ease of drawing a lens
has been added to the diagram in place of a
very long distance, this is so we can assume
that the rays are parallel as they hit the
first order maximum.
In order to get constructive interference, the path difference between adjacent rays must
be an integer number of wavelengths. This means that the second ray down is λ longer, the
third is 2λ longer etc.
Using the small red triangle on the diagram we can calculate λ using the angle θ and the
separation of the slits. However, θ shown on the red triangle would be very difficult to
measure. It’s therefore lucky that the rules of geometry state that the angle between point
P, the centre line on the grating and the centre line on the screen is also θ. This we can
measure.
Purely from our geometrical relationships, we can state that:
λ = d sinθ
For a second order maximum, the path difference is 2λ, so
2λ = d sinθ
And in general:
nλ = d sinθ
Stationary (Standing) Waves
We previously used graphical methods to show the superposition of two waves moving at the
same speed in the same direction. What happens if the waves move in opposite directions?
Exercise
1. Using graphical methods (plot it on graph paper) plot the superposition of two
coherent waves that are 180o out of phase.
2. Using graphical methods (plot it on graph paper) plot the superposition of two
coherent waves that are exactly in phase.
Now imagine the waves are moving past each other in opposite directions. The resultant wave
will constantly oscillate between the two states you’ve plotted.
[Software Demo, VPL – stationary waves & http://en.wikipedia.org/wiki/Standing_wave]
Formation of a Stationary Wave
A stationary wave results when 2 waves which are travelling in opposite directions and which
have the same speed and wavelength and approximately equal amplitudes are superposed.
This diagram shows a number of standing waves. The thick
black line shows the maximum deflection of the wave, and
the thinner ones show intermediate stages.
Stationary Wave Properties
Stationary (or standing, two words for the same thing)
waves don’t go anywhere; they simply oscillate up and
down in the same place.
A point on a stationary wave that does not move up and
down at all is called a node.
A point on a stationary wave that experiences maximum
deflection from the equilibrium position is called an antinode
Nodes
Nodes
Stationary waves still have an amplitude, wavelength and
frequency just like a progressive wave, their speed is taken
to be the speed of one of the two waves travelling in
opposite directions that make up the stationary wave.
Separation between adjacent nodes (or antinodes) = λ/2
Anitnodes
Which can clearly be seen from the diagrams.
Demonstrating Stationary Waves
The simplest way to demonstrate a stationary wave is with a spring. You simply anchor one
end to something solid, hold the other end and oscillate it up and down.
[stationary wave on a rope - http://www.ngsir.netfirms.com/englishhtm/StatWave.htm]
The frequency of oscillation of the spring will depend on three factors:
• The length of the spring;
• The tension of the spring;
• The mass per unit length of the spring.
We can prove this using the sonometer.
[Sonometer experiment – include strobe light to see the movement of the wire]
Microwave Ovens
A stationary electromagnetic wave is generated by a microwave oven. It is this that causes
the water molecules in food to switch polarity extremely quickly. When the water molecules
spin round they bang into other nearby molecules, which creates heating (heat is simply jiggly
molecules after all).
You can prove this by removing the turntable from a microwave oven and placing something
meltable like a bar of chocolate inside. After a few seconds you will see melted patches and
solid patches of chocolate all the way along, showing where the nodes and antinodes occur.
How could you measure the speed of light using this method?
Stationary Waves in Music
By sending a pressure wave down a pipe it is possible to produce a standing wave in the air
column within the pipe. This is how many musical instruments work. Remember that sound is a
longitudinal wave, so it’s quite possible to produce stationary longitudinal waves as well as
transverse ones.
[Experiment – resonance tubes, shown at end of notes]
The wave reflects back from the end of the pipe, and since we then have two identical waves
moving in opposite directions, a standing wave is produced. This often produces a sound that
we can hear, such as when you blow over the top of a milk bottle, or blow a flute, or vibrate a
reed in a clarinet or saxophone, or vibrate a guitar string over the hollow body of an acoustic
guitar.......
The diagram on the right shows where the nodes and
antinodes lie on standing waves in pipes with different
configurations of open and closed ends. There is always a
node at the open end of a pipe, and an antinode at the closed
end.
Fundamental Frequency – on a string
The fundamental frequency of a standing wave on a string is
the lowest frequency at which a particular string will support
a standing wave. If you pluck a guitar string, this is the
lowest note you hear.
If both ends of the string are anchored, then ½ a wavelength
will fit onto the string.
Figure - Fundamental frequency in
pipes
Harmonics
A Harmonic is a multiple of the fundamental
frequency. The fundamental frequency is called
the 1st harmonic, the next highest frequency is
the 2nd harmonic and so on.
On any plucked string a combination of harmonics
is produced. It is the amplitude of each harmonic
that gives the instrument its timbre (unique
sound). This is why a guitar sounds different to a
violin.
Harmonics and Fundamental Frequency
Frequen
in Pipes
If you look at the fundamental frequency in pipes figure above, you will see that when a
stationary wave vibrating at the fundamental frequency is produced in a pipe the number of
wavelengths inside the pipe changes depending on whether the ends
ends are open or not. For a
pipe with two open ends or two closed ends half a wavelength fits in, but for one with one
open end and one closed end you only get ¼ of a wavelength inside.
Most wind instruments are of this orientation, trumpets, clarinets, saxophones,
saxop
trombones
etc. The exception is the flute, which has two open ends.
For those who wish to understand this more fully, this website is brilliant:
http://www.soundonsound.com/sos/apr01/articles/synthsecrets.asp
/sos/apr01/articles/synthsecrets.asp
Exercise –
1. Draw the 2nd, 3rd and 4th harmonics in a pipe with one end open and one end closed.
2. Draw the 2nd, 3rd and 4th harmonics of a vibrating string with two fixed ends.
End Correction
In reality, when a standing
ng wave is produced in a pipe with one end open and one end closed,
the frequency of the note produced is lower than expected. This is because there is a small
volume of air outside the open end of the pipe that also takes part in the vibrations,
effectively
ly giving the pipe extra length. When calculations are carried out, this must be
taken into account.
Measuring the Speed of Sound in Air Using a Tube Open at One
End
This is a relatively simple experiment and uses only your
knowledge of the wave equation (v = fλ) and of how
stationary waves form in a pipe:
•
Hold a tuning fork over one end of the pipe;
•
Starting from 0 length, lengthen the pipe until you
hear a note;
•
This note will be at the fundamental frequency;
•
Use your knowledge of the number of wavelengths
in this type of pipe at the fundamental frequency and the
wave equation v = fλ to calculate the speed of sound in air;
•
Frequency is simply read from the tuning fork;
•
Don’t forget the end correction!